Complete arcs in inversive planes over prime fields (Q1849891)

In the paper under review some new infinite families of complete arcs in Miquelian inversive planes of large prime order are constructed. Let \({\mathcal M}_q\) be the inversive Miquelian plane, where \(q = p^h\) and \(p\) is an odd integer prime. An arc of size \(k\) (briefly, a \(k\)-arc) of \({\mathcal M}_q\) is a set of \(k\)-points no four of which lie on the same circle. An arc is complete if it is not contained in a larger arc of \({\mathcal M}_q\). The existence problem for an infinite family of complete arcs in Miquelian inversive planes has been solved by the latter author in [\textit{A. Sonnino}, J. Geom. 66, No. 1-2, 187-191 (1999; Zbl 0947.51008)], where a complete \(\frac{1}{3}(q + 5)\)-arc in \({\mathcal M}_q\), \(q\) odd is constructed. In the paper under review the authors prove the following two theorems which give some other infinite families of complete arcs in Miquelian planes \({\mathcal M}_p\), where \(p\) is a large integer prime. Theorem 1. For any prime integer \(p\geq 1.24\times 10^8\) the Miquelian inversive plane \({\mathcal M}_p\) contains a complete \(((p + 17)/4)\)-arc. \textbf{Theorem 2.} Let \(p\) be any prime integer, \(p\geq 5.8\times 10^{10}\). Then the Miquelian inversive plane \({\mathcal M}_p\) contains complete \((m + 3 + k)\)-arcs for all \(m\) such that \((p + 2)/4 > m > (p - 6)/14\) and \(k = (p + 8)/(2 m + 2)- 2\).